Eigenvalues of the Laplacian in a Disk with the Dirichlet Condition on Finitely Many Small Boundary Parts in the Critical Case
- Autores: Gadyl’shin R.1,2, Rep’evskii S.3, Shishkina E.1
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Afiliações:
- M. Akmullah Bashkir State Pedagogical University
- Bashkir State University
- Chelyabinsk State University
- Edição: Volume 213, Nº 4 (2016)
- Páginas: 510-529
- Seção: Article
- URL: https://journals.rcsi.science/1072-3374/article/view/237200
- DOI: https://doi.org/10.1007/s10958-016-2722-4
- ID: 237200
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Resumo
We consider the boundary value problem for eigenvalues of the negative Laplace operator in a disk with the Neumann boundary condition on the circle except for finitely many (more than 1) small arcs, where the Dirichlet boundary condition is imposed, with lengths tending to zero. We construct complete asymptotics expansions of egenvalues with respect to the parameter (the arc length) converging to a double eigenvalue to the limit Neumann problem, in the critical case, where one of the eigenfunctions of the limit problem vanishes at all contraction points for small arcs.
Sobre autores
R. Gadyl’shin
M. Akmullah Bashkir State Pedagogical University; Bashkir State University
Autor responsável pela correspondência
Email: gadylshin@yandex.ru
Rússia, 3a, Oktyabrskoy Revolutsii St., Ufa, 450000; 32, Frunze st., Ufa, 450074
S. Rep’evskii
Chelyabinsk State University
Email: gadylshin@yandex.ru
Rússia, 129, Brat’ev Kashirinykh St., Chelyabinsk, 454000
E. Shishkina
M. Akmullah Bashkir State Pedagogical University
Email: gadylshin@yandex.ru
Rússia, 3a, Oktyabrskoy Revolutsii St., Ufa, 450000